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{{short description|One of several theorems linking the sizes of different ideal class groups}}
{{Confusing|article|date=February 2010}}
In [[algebraic number theory]], a '''reflection theorem''' or '''Spiegelungssatz''' ([[German language|German]] for ''reflection theorem'' – see ''[[Spiegel (disambiguation)|Spiegel]]'' and ''[[Satz (disambiguation)|Satz]]'') is one of a collection of theorems linking the sizes of different [[ideal class group]]s (or [[ray class group]]s), or the sizes of different [[isotypic component]]s of a class group. The original example is due to [[Ernst Kummer|Ernst Eduard Kummer]], who showed that the class number of the [[cyclotomic field]] <math>\mathbb{Q} \left( \zeta_p \right)</math>, with ''p'' a prime number, will be divisible by ''p'' if the class number of the maximal real subfield <math>\mathbb{Q} \left( \zeta_p \right)^{+}</math> is. Another example is due to Scholz.<ref>A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, ''J. reine angew. Math.'', '''166''' (1932), 201-203.</ref> A simplified version of his theorem states that if 3 divides the class number of a [[real quadratic field]] <math>\mathbb{Q} \left( \sqrt{d} \right)</math>, then 3 also divides the class number of the [[imaginary quadratic field]] <math>\mathbb{Q} \left( \sqrt{-3d} \right)</math>.▼
▲:''For reflection principles in set theory, see [[reflection principle]].''
▲In [[algebraic number theory]], a '''reflection theorem''' or '''Spiegelungssatz''' ([[German language|German]] for ''reflection theorem'' – see ''[[Spiegel]]'' and ''[[Satz (disambiguation)|Satz]]'') is one of a collection of theorems linking the sizes of different [[ideal class group]]s (or [[ray class group]]s), or the sizes of different [[isotypic component]]s of a class group. The original example is due to [[Ernst Kummer|Ernst Eduard Kummer]], who showed that the class number of the [[cyclotomic field]] <math>\mathbb{Q} \left( \zeta_p \right)</math>, with ''p'' a prime number, will be divisible by ''p'' if the class number of the maximal real subfield <math>\mathbb{Q} \left( \zeta_p \right)^{+}</math> is. Another example is due to Scholz.<ref>A. Scholz, Uber die Beziehung der Klassenzahlen quadratischer Korper zueinander, ''J. reine angew. Math.'', '''166''' (1932), 201-203.</ref> A simplified version of his theorem states that if 3 divides the class number of a [[real quadratic field]] <math>\mathbb{Q} \left( \sqrt{d} \right)</math>, then 3 also divides the class number of the [[imaginary quadratic field]] <math>\mathbb{Q} \left( \sqrt{-3d} \right)</math>.
==Leopoldt's Spiegelungssatz==
Both of the above results are generalized by [[Heinrich-Wolfgang Leopoldt|Leopoldt]]'s "Spiegelungssatz", which relates the [[p-rank]]s of different isotypic components of the class group of a number field considered as a [[Galois module|module]] over the [[Galois group]] of a Galois extension.
Let ''L''/''K'' be a finite Galois extension of number fields, with group ''G'', degree prime to ''p'' and ''L'' containing the ''p''-th roots of unity. Let ''A'' be the ''p''-Sylow subgroup of the class group of ''L''. Let φ run over the irreducible characters of the group ring '''Q'''<sub>''p''</sub>[''G''] and let ''A''<sub>φ</sub> denote the corresponding direct summands of ''A''. For any φ let ''q'' = ''p''<sup>φ(1)</
:<math> [ A_\phi : A_\phi^p ] = q^{e_\phi} . </math>
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:<math> \phi^*(g) = \omega(g) \phi(g^{-1}) . </math>
Let ''E'' be the unit group of ''K''. We say that ε is "primary" if <math>K(\sqrt[p]\epsilon)/K</math> is unramified, and let ''E''<sub>0</sub> denote the group of primary units modulo ''E''<sup>''p''</sup>. Let δ<sub>φ</sub> denote the ''G''-rank of the φ component of ''E''<sub>0</sub>.
The Spiegelungssatz states that
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